Linear Algebra and Its Applications, Exercise 2.6.3

Exercise 2.6.3. Suppose we form the product of  2 by 2 matrices representing 5 reflections and 8 rotations. Does that product matrix represent a reflection or a rotation?

Answer: I’ll show a long way to the answer and then a shorter (and more general) way. First, the long way:

Let Q_1 through Q_8 be the 8 rotation matrices and H_1 through H_5 the 5 reflection matrices. Note that the question is a bit ambiguous as to whether the reflections or rotations are done first; we assume that the reflections are done first, so that the product matrix A is calculated as

A = Q_8 Q_7 Q_6 Q_5 Q_4 Q_3 Q_2 Q_1 H_5 H_4 H_3 H_2 H_1

(As I discuss below regarding the simpler way to an answer, whether the reflections come first or the rotations doesn’t actually matter.)

Since matrix multiplication is associative we can do the multiplications as follows:

A = (Q_8 Q_7) (Q_6 Q_5) (Q_4 Q_3) (Q_2 Q_1) H_5 (H_4 H_3) (H_2 H_1)

From the previous post on composing reflections and rotations we know that the product of two rotations is a rotation, and the product of two reflections is also a rotation. The above equation thus reduces to

A = Q_9 Q_{10} Q_{11} Q_{12} H_5 Q_{13} Q_{14}

where Q_9 is a rotation matrix formed by the product of Q_8 and Q_7, Q_{14} is a rotation matrix formed by the product of H_2 and H_1, and Q_{10} through Q_{13} are calculated similarly.

We can further reduce this as follows:

A = (Q_9 Q_{10}) (Q_{11} Q_{12}) H_5 (Q_{13} Q_{14}) = Q_{15} H_5 Q_{16}

where Q_{15} is a rotation matrix formed by the product of Q_{9} and Q_{10} and Q_{16} is the rotation matrix formed by the product of Q_{13} and Q_{14}.

Finally, we know from the previous post that the product of a reflection and a rotation (or a rotation and a reflection) is a reflection, so we can reduce the previous equation to

A = (Q_{15} H_5) Q_{12} = H_6 Q_{16} = H_7

where H_{6} is a reflection matrix formed by the product of Q_{15} and H_5 and H_{7} is a reflection matrix formed by the product of H_6 and Q_{16}. So the product of 5 reflections and 8 rotations is a reflection.

Now for the shorter way, which can be used no matter the number of rotations and reflections:

Note first that the product of any number of rotation matrices will itself be a rotation matrix. This follows straightforwardly by induction from the fact that the product of two rotation matrices is a rotation matrix.

Then note that the product of an even number of reflection matrices will be a rotation matrix. This is because we can pair up the reflection matrices, multiply the two matrices in each pair to produce a rotation matrix, and then multiply the resulting rotation matrices (no matter how many there are) to produce a rotation matrix.

Finally, note that the product of an odd number of reflection matrices will be a reflection matrix: We can take all reflection matrices except the last one and multiply them; since there will an even number of reflection matrices this will produce a rotation matrix, as discussed in the previous paragraph. We then take this rotation matrix and multiply it by the remaining reflection matrix to produce a reflection matrix.

So, if we have the product of 5 reflections and 8 rotations, the 8 rotations will produce a rotation, the 5 reflections will produce a reflection (since 5 is odd), and the product of the resulting rotation and reflection matrices will produce a reflection.

Note that it is irrelevant whether the rotations are done first or the reflections. In either case the 8 rotations will reduce to a rotation and the 5 reflections will reduce to a reflection, and the product of a rotation and a reflection is a reflection no matter in which order the matrices are multiplied.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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